Big dimensionless numbers (in Planks units)

[Dmitry67]

Recently I found this:

The formula for power output in Watts by Hawking radiation for a simple non-rotating hole of mass M kg is
$$\frac{\hbar c^6}{15360 \pi G^2} M^{-2}$$

Note the number - 15360*pi = 48254...
This is a biggest number in Planks units I have ever seen

In Planks units, what is the biggest dimensionless you have even seen?
Anything bigger than 50'000?

 

[Demystifier]

Inverse cosmological constant. It is a number with 120 digits. :tongue:

But you probably meant a number which can be calculated from first principles, right?

 

[MTd2]

Avogadro constant.
Edit.:
(damn, just saw plank units)

 

[Demystifier]

Avogadro constant.

My is bigger. :tongue:

 

[Dmitry67]

But you probably meant a number which can be calculated from first principles, right?

Yes, correct.
So current bid is 48254
One... Two... Anyone?

 

[sylas]

Yes, correct.
So current bid is 48254
One... Two... Anyone?

Well, $\hbar$ is just $h/2\pi$, so that formula, which is the instantaneous power output of a simple non-rotating black hole of mass M, is also written

$$\frac{h c^6}{30720 \pi^2 G^2} M^{-2}$$​

Note that the qualification "Planck Units" makes no sense. The number is dimensionless. It will apply for any set of units. What changes with the units are the values for G, h and c. The additional dimensionless factor, if we use h rather than hbar, is about 303194.2472

Also, what I found interesting about that formula was the index of c. c is generally pretty large, unless you pick units to make it 1. And this raises it to the sixth power! Off the top of my head I cannot think of another natural physical relation which raises something to fixed powers more than 6.

 

[Demystifier]

The number of string vacua is estimated to be something like 10^1000 or bigger. Even though it is not calculated exactly, it is calculated from first principles and corresponds to the number of different topologies of Calabi-Yau spaces describing possible compactifications of the additional dimensions in string theory.

Or should we only count numbers in 3+1 dimensional physics?

 

[MTd2]

Well, that should be something related to plank units.

 

[Demystifier]

In pure mathematics we also have this
http://en.wikipedia.org/wiki/Monster_group
very big number, although I am not sure if it has anything to do with physics.

 

[Demystifier]

Well, that should be something related to plank units.

See #6.

 

[Dmitry67]

Here is my turn.
Lets say we have a black hole of mass M
We wait until it evaporates completely.
So the original BH it is replaced with a sphere of hawking radiation
That Hawking radiation occupies much more space than before.

Evaporation time:

$$t_e = 5120 \pi M^3$$

Radius of the sphere is the same.
Before it was

$$r_s = 2 M$$

$$\frac{r_e}{r_s} = 2560 \pi M^2$$

and for the volume

$$\frac{V_e}{V_s} = 16777216000 \pi^3 M^6$$

so the constant is 520199001176

 

[arivero]

Just to repeat the precision done above: a dimensionless number is dimensionless, it does not relate to the units. It is usually a quotient of two measures with the same dimensions, for instance the fine structure is a quotient of two angular momenta.

 

[Dmitry67]

yes, but in Planks units their nature is more visible.

 

[thubsch]

The number of string vacua is estimated to be something like 10^1000 or bigger. Even though it is not calculated exactly, it is calculated from first principles and corresponds to the number of different topologies of Calabi-Yau spaces describing possible compactifications of the additional dimensions in string theory.

Or should we only count numbers in 3+1 dimensional physics?

Unh... Here goes: well-nigh every (super)string compactification thus far constructed ends up having at least one continuous parameter, and so the "number" of such particular compactifications is uncountable. Moreover, typical Calabi-Yau compactifications ever constructed have many (tens, hundreds, some even a thousand or so) continuous parameters. And, that's not all: Miles Reid (a mathematician of considerable repute in the field) is on record having conjectured that there may well be possible to construct indefinite sequences of different "topological types" of Calabi-Yau manifolds, each one suc manifold equipped with an ever-larger-dimensional parameter space.

In turn, the physics of such compactifications imposes a certain quantization effect (discovered as best as I know by Joe Polchinski of the Kavli Institute, Santa Barbara), whereby in this vast continuum of Calabi-Yau (and related) compactifications, only a "sufficiently dense" subset represents completely consistent models. This quantized subset is what is sometimes guestimated as 10^500 (give or take a Googol ), and is called "discretuum".